Tuesday, August 20, 2013

VIX Futures & Options are one of the most actively traded index
derivatives series on the Chicago Board Options Exchange (CBOE). These
derivatives are written on S&P 500 volatility index and their popularity
has made volatility a widely accepted asset class for trading, diversifying and hedging
instrument since their launch. VIX Futures
started trading on March 26th, 2004 on CFE (CBOE Future Exchange)
and VIX Options were introduced on Feb 24th, 2006.

VIX Futures & Options

VIX (Volatility Index) or the ‘Fear Index’ is based on the S&P 500
options volatility. Spot VIX can be defined as square root of 30 day variance
swap of S&P 500 index (SPX) or in simple terms it is the 30-day average
implied volatility of S&P 500 index options. The VIX F&O are based on
this spot VIX and is similar to the equity indexes in general modus operandi.
But structurally they have far more differences than similarities. While, in
case of equity indices (for example SPX), the index is a weighted average of
the components, in case of the VIX it is sum of squares of the components. This
non-linear relationship makes the spot VIX non-tradable but at the same time
the derivatives of spot VIX are tradable. This can be better understood with
the analogy of Interest Rate Derivatives. The derivatives based on the interest
rates are traded worldwide but the underlying asset: interest rate itself
cannot be traded.

The different relation between the VIX derivatives and the underlying
VIX makes it unique in the sense that the overall behavior of the instruments
and their pricing is quite different from the equity index derivatives. This
also makes the pricing of VIX F&O a complicated process. A proper
statistical approach incorporating the various aspects like the strength of
trend, mean reversion and volatility etc. is needed for modeling the pricing
and behavior of VIX derivatives.

Research on Pricing Models

There has been a lot of research in deriving models for the VIX F&O
pricing based on different approaches. These models have their own merits and
demerits and it becomes a tough decision to decide on the most optimum model.
In this regards, I find the work of Mr.
Qunfang Bao titled‘Mean-Reverting
Logarithmic Modeling of VIX’quite interesting. In his research,
Bao not only revisits the existing models and work by other prominent
researchers but also comes out with suggestive models after a careful
observation of the limitations of the already proposed models. The basic thesis
of Bao’s work involves mean-reverting logarithmic dynamics as an essential
aspect of Spot VIX.

VIX F&O contracts don’t necessarily track the underlying in the same
way in which equity futures track their indices. VIX Futures have a dynamic
relationship with the VIX index and do not exactly follow its index. This
correlation is weaker and evolves over time. Close to expiration, the
correlation improves and the futures might move in sync with the index. On the
other hand VIX Options are more related to the futures and can be priced off
the VIX futures in a much better way than the VIX index itself.

Pricing Models

As a volatility index, VIX shares the properties of mean reversion,
large upward jumps & stochastic volatility (aka stochastic vol-of-vol). A good model is expected to take into
consideration, most of these factors.

There are roughly two categories of approaches for VIX modeling. One is
the Consistent approach and the other being Standalone approach.

I. Consistent Approach: - This is the pure diffusion model wherein the inherent relationship
between S&P 500 & VIX is used in deriving the expression for spot VIX
which by definition is square root of forward realized variance of SPX.

II. Standalone
Approach: - In this approach, the VIX
dynamics are directly specified and thus the VIX derivatives can be priced in a
much simpler way. This approach only focuses on pricing derivatives written on
VIX index without considering SPX option.

Bao in his paper mentions that the standalone approach is comparatively
better and simpler than the consistent approach.

MRLR model

The most widely proposed model under the standalone approach is MRLR
(Mean Reverting Logarithmic Model) model which assumes that the spot VIX
follows a Geometric Brownian motion process. The MRLR model fits well for VIX
Future pricing but appears to be unsuited for the VIX Options pricing because
of the fact that this model generates no skew for VIX option. In contrast, this
model is a good model for VIX futures.

MRLRJ model

Since the MRLR model is unable to produce implied volatility skew for
VIX options, Bao further tries to modify the MRLR model by adding jump into the
mean reverting logarithmic dynamics obtaining the Mean Reverting Logarithmic
Jump Model (MRLRJ). By adding upward jump into spot VIX, this model is able to
capture the positive skew observed in VIX options market.

MRLRSV model

Another way in which the implied volatility skew can be produced for VIX
Options is by including stochastic volatility into the spot VIX dynamics. This
model of Mean Reverting Logarithmic model with stochastic volatility (MRLRSV)
is based on the aforesaid process of skew appropriation.

Bao further combines the MRLRJ and MRLRSV models together to form
MRLRSVJ model. He mentions that this combined model becomes somewhat
complicated and in return adds little value to the MRLRJ or MRLRSV models. Also
extra parameters are needed to be estimated in case of MRLRSVJ model.

MRLRJ
& MRLRSV models serve better than the other models that have been proposed
for pricing the VIX F&O. Bao in his paper, additionally derives and
calibrates the mathematical expressions for the models he proposes and derives
the hedging strategies based on these models as well. Quantifying the
Volatility skew has been an active area of interest for researchers and this
research paper addresses the same in a very scientific way, keeping in view the
convexity adjustments, future correlation and numerical analysis of the models
etc. While further validation and back testing of the models may be required,
but Bao’s work definitely answers a lot of anomalous features of the VIX and
its derivatives.---Azouz Gmach works for QuantShare, a technical/fundamental analysis software.===My online Mean Reversion Strategies workshop will be offered in September. Please visit epchan.com/my-workshops for registration details.